Can someone help me to understand the alpha-beta pruning algorithm?
Suppose that you have already search a part of the complete search tree, for example the complete left half. This may not yet give you the true game-theoretic value for the root node, but it can already give you some bounds on the game-theoretic value that the player to play in the root node (let's say, the max player) can guarantee by moving into that part of the search tree. Those bounds / guarantees are:


$\alpha$: the minimum score that the maximizing player already knows it can guarantee if we move into the part of the search tree searched so far. Maybe it can still do better (get higher values) by moving into the unsearched part, but it can already definitely get this value.
$\beta$: the maximum score that the minimizing player already knows it can guarantee if we moves into the part of the search tree searched so far. Maybe it can still do better (get lower values) by moving into the unsearched part, but it can already definitely get this value.


The intuitive idea behind alpha-beta pruning is to prune chunks of the search tree that become uninteresting for either player because they already know they can guarantee better based on the $\alpha$ or $\beta$ bounds.



For a simple example, suppose $\alpha = 1$, which means that the maximizing player already has explored a part of the search tree such that it can guarantee at least a value of $1$ by playing inside that part (the minimizing player has no options inside that entire tree to reduce the value below $1$, if the maximizing player plays optimally in that part).

Suppose that, in the current search process, we have arrived at a node where the minimizing player is to play, and it has a long list of child nodes. We evaluate the first of those children, and find a value of $0$. This means that, under the assumption that we reach this node, the minimizing player can already guarantee a value of $0$ (and possibly get even lower, we didn't evaluate the other children yet). But this is worse (for the maximizing player) than the $\alpha = 1$ bound we already had. Without evaluating any of the other children, we can already tell that this part of the search tree is uninteresting, that the maximizing player would make sure that we never end up here, so we can prune the remaining children (which could each have large subtrees below them).